The Problem Solving Methodology is a tool to help you improve your proficiency at the process of problem solving. Without a methodology, most people are more easily lost and ineffective when it comes to solving difficult problems. Having a common methodology to use in problem solving contexts increases your confidence and the quality of solutions and decisions you make.
| 1. Define the Problem | Identify and clearly state the problem |
| 2. Identify the Key Issues | Determine important issues associated with the problem |
| 3. Collect and Assess Information | Collect and assess available information relevant to the problem; determine what information is missing |
| 4. Identify Assumptions | Clarify what assumptions are being made about the problem |
| 5. Break Problem Apart | Separate the problem into smaller sub-problems |
| 6. Model Sub-problems | Generate solutions for each sub-problem |
| 7. Integrate Solutions | Integrate the solutions from sub-problems into main problem |
| 8. Test and Validate | Validate the solution; assess the quality of the solution |
| 9. Generalize the Solution | Determine how to generalize the problem solution for use in other situations |
| 10. Communicate the Solution | Present to others in oral and/or written form the solution and documentation of the process |
You have completed your first semester at college and are returning home for the Christmas holidays. Since you will be home for five or six weeks, you want to get a short- term job to keep you busy and earn spending money for when you return to school.
Finding and choosing a job during the holiday break.
Must be a short-term job, maximum money, and within commuting distance from home.
Obtain a list of temporary agencies from the phone book and the names of stores in nearby malls.
Retail businesses need short-term, full-time jobs in December, and employers reward initiative.
Who is hiring for the most money? Who is hiring for the most hours per week? How many weeks will the employment last?
Amount of money = number of hours/week * hourly wage * number of weeks worked.
The temporary agency can get you work for 5 weeks @ 40 hours/week for $7.00/hour: a possible total of $1400. This is if they are satisfied with your work evaluations. If the evaluations are not satisfactory, you may get no other jobs from the agency.
Other than the temporary agency, the best paying job is for $10.00/hour for 3 weeks (total $450). The job with the greatest number of hours is 40 hours/week for six weeks at $5.50/ hour (total $1320). You choose the temporary agency feeling that your evaluations will be strong and you will get the full-time work for five weeks.
You call all businesses to verify your data.
You generalize that the temporary agency is a good solution for, if successful, you can also work there over spring break.
You write a letter to the agency accepting its job offer and explaining the process you used to make your decision.
The first step in the problem solving process is to correctly and clearly define the problem. The ability to assess situations and interpret information properly contributes to correct problem definitions. It is important to define the problem clearly and correctly so that time and effort are not expended in an activity that leads to the solution of the wrong problem. Sometimes it is helpful to get the opinion of others who may perceive and see problem situations differently than you. Their input may improve your original problem statement.
By asking critical questions, you can identify and clarify important issues surrounding the problem which should be considered as you work through the problem solving process. Diagrams associated with the problem situation can also help to identify key issues. When key issues are identified, it often leads to stronger and more comprehensive problem solutions. Sometimes the key issues may cause you to reformulate the problem definition.
Once you have defined the problem and understand what the key issues are, you have a better idea as to what information is most relevant and necessary to solve the problem. Assess the quality of the information you collect based on criteria such as accuracy, reliability, appropriateness, and accessibility. Strong information processing and assessment skills produce better quality information resources that contribute to clearer insights and more creative solutions.
Before proceeding any further in the methodology, you should identify and write down the assumptions you are making concerning the problem situation. Be sure to test the validity of the assumptions you make. The problem definition, the scope of the issues raised, and the quantity and quality of the information you collect all influence the assumptions you make. For example, if you have less available information, you may have to make more assumptions. In some cases, you want to make valid assumptions to help simplify the process of solving the problem. Finally, realize that when you change, alter, or make additional assumptions, it can lead to completely new and different solutions. If the assumption is false, it can lead to wrong or poor quality solutions.
In general, the problem solving process is made more effective and efficient by dividing the problem into manageable, logical pieces or sub-problems, dealt with one at a time. Subdividing or breaking apart the problem makes it easier to begin developing and formulating possible solutions. With complex problems, breaking apart the problem is a necessity.
Once the problem has been broken apart, you must generate possible solutions to the subproblems. Building models that replicate the principles and relationships at work in a given problem can be a great help with the sub-problems. Examples of models include diagrams, equations, graphs, tables, and computer programs. Models should make use of available and appropriate resources, including the information and assumptions from steps 3 & 4 and your own knowledge, experiences, and creativity. Realize that many times, there is not one right answer. Therefore, you should generate several possible solutions which you later evaluate.
The solutions to the sub-problems generated in the previous step must be put together, in many cases with modifications. This involves evaluating possible solutions and determining how the parts will best work as a whole. The result is often a set of larger models, which serve as possible solutions or means to a solution for the defined problem. The next step of the methodology requires you the examine and assess these solutions.
Since the outcome of step 7 of the methodology typically results in more than one solution, criteria need to be established to assess the possible solutions. The testing and validating process involves using these criteria to determine how well each possible solution measures up. The strength of the assumptions should also be tested because the choice and ranking of solutions may vary based on the assumptions made.
A solution to a problem becomes much more valuable and useful when it can be generalized and applied to many different situations rather than being limited to one unique situation. By making modifications and adaptations to the solution, you can make it such that the solution will work for others as well as for yourself. Finally, you can save yourself a great deal of time and effort in the future by using previously solved, generalized problem solutions to applicable situations.
In many situations, you must communicate your solution(s) and the processes used to arrive at the solution(s) to an audience. It is important that solutions to problems be effectively and persuasively communicated. Otherwise, the value of the solution and all the work associated with it are diminished or even dismissed. You want your oral and/or written communication to include the significance of the problem, the fact that assumptions have been made and tested, that you have examined possible solutions, and your final recommendations and conclusions.
Furnco manufactures tables and chairs. The company wants to produce the number of tables and chairs which will result in the highest profit given the limited amount of raw materials and labor available.
The problem stated in the form of a question is, how many tables and how many chairs should Furnco manufacture in order to maximize profit.
We need to let x represent the number of chairs to be produced and y to represent the number of tables to be produced. We then know that
(1) 20x + 40y <= 40000
1000|\
|-\
|--\
|---\
|----\
|-----\
|------\
____|_______\______________________
0 2000
since each chair takes 20 board feet and each table takes 40 board feet and there are 40,000 board feet available.
(2) 4x + 3y <= 6000.
2000|
|\
|-\
|--\
|---\
|----\
|-----\
|------\
____|_______\_________
0 1500
since each chair requires 4 hours and each table requires 3 hours of labor and there are 6,000 hours of labor available.
We know that the wood will cost
(3) 20x + 40y
since each board foot of wood costs $1 and the labor will cost
(4) 15(4x + 3y).
since each hour of labor costs $15
The sales value will be
(5) 110x + 135y.
since chairs sell for $110 and tables for $135.
The profit for Furnco is
(6) 110x + 135y - (20x + 40y + 15(4x + 3y))
(7) = 30x + 50y
Putting the two graphs (1) \ and (2) * together and plotting the line o 30x + 50y = 50,000 we obtain
2000|
|\
| \
| \
| \
| \
| \
1000|o \
|* o \
|---*o \
|------* \
|--------o*\
|----------o\*
|------------o *
____|_____________\o___*_______
0 1 1 2
5 6 0
0 6 0
0 7 0
From the above graph, it is clear that the maximum profit will occur when the profit line passes through the intersection of the lines that bound the shaded region. We find this intersection point by solving the equations 20x + 40y = 40000 and 4x +3y = 6000 simultaneously.
(-2) 2x + 4y = 4000
4x + 3y = 6000
-5y = -2000
y = 400
x = 1200
Thus, Furnco should make 400 tables and 1200 chairs to obtain a maximum profit of $186,000.
Furnco checks the total board feet of wood needed to make 400 tables and 1200 chairs and observes that the total is 20(1200) + 40(400) = 40,000. They check that the total hours of labor would be 4(1200) + 3(400) = 6,000. The owner verifies that this production ratio agrees with customer demand.
Furnco's owner generalizes that the process used in this problem can be used when the availability of material and labor changes, especially the graphical representations. He can see that a production schedule that maximizes profit may not always meet customer need. For example, suppose most customers want more than three chairs with each table.
Furnco's owner writes up the solution and the process used in reaching it. He makes this report available to his foreman so the production schedule can be correctly organized. He uses the profit figure to help him plan the tax consequences of the business and do his estate planning for the year.